AP Calculus AB 2010 Scoring Guidelines

1 AP Calculus AB 2010 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the College Board is composed of more than 5,700 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,800 colleges through major programs and services in college readiness, college admission, guidance, assessment, financial aid and enrollment.

2 Among its widely recognized programs are the SAT , the PSAT/NMSQT , the Advanced Placement Program (AP ), SpringBoard and ACCUPLACER . The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities and concerns. 2010 The College Board. College Board, ACCUPLACER, Advanced Placement Program, AP, AP Central, SAT, SpringBoard and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service is a trademark owned by the College Board.

3 PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: Visit the College Board on the Web: AP Central is the official online home for the AP Program: AP Calculus AB 2010 Scoring Guidelines Question 1 2010 The College Board. Visit the College Board on the Web: There is no snow on Janet s driveway when snow begins to fall at midnight.

4 From midnight to 9 , snow accumulates on the driveway at a rate modeled by ()cos7tftte= cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 () The rate (),gt in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by ()0for0 6125 for 67108 for 79 .tgttt < = < (a) How many cubic feet of snow have accumulated on the driveway by 6 (b) Find the rate of change of the volume of snow on the driveway at 8 (c) Let ()ht represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight.

5 Express h as a piecewise-defined function with domain (d) How many cubic feet of snow are on the driveway at 9 (a) () or cubic feet 2 : { 1 : integral 1 : answer (b) Rate of change is ()() = or cubic feet per hour. 1 : answer (c) ()00h= For 06,t< ()( )( ) +=+= For 67,t< ()( )( )()66601251256 .tththgsdsdst=+=+= For 79,t< ()( )( )()7712510781251087 .ttgsdsdshtht=+=+=+ Thus, ()()()0for061256for 671251087for 79thttttt = < + < 3 : ()()() 1 : for 06 1 : for 67 1 : for 79htthtthtt < < (d) Amount of snow is ()( ) = or cubic feet.}

6 3 : () 1 : integral 1 : 9 1 : answerh AP Calculus AB 2010 Scoring Guidelines Question 2 2010 The College Board. Visit the College Board on the Web: A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon ()0t= and 8 () The number of entries in the box t hours after noon is modeled by a differentiable function E for Values of (),Et in hundreds of entries, at various times t are shown in the table above.

7 (a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time Show the computations that lead to your answer. (b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of () Using correct units, explain the meaning of ()8018 Etdt in terms of the number of entries. (c) At 8 , volunteers began to process the entries. They processed the entries at a rate modeled by the function P, where ()3230298976 Ptttt= + hundreds of entries per hour for According to the model, how many entries had not yet been processed by midnight ()12 ?

8 T= (d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer. (a) ()()()756475 EEE = hundred entries per hour 1 : answer (b) ()()()()()()()()()80022557 7812 3 2 1 or 10 ++++ +++ = ()8018 Etdt is the average number of hundreds of entries in the box between noon and 8 3 : 1 : trapezoidal sum 1 : approximation 1 : meaning (c) ()1282323 167 Ptdt = = hundred entries 2 : { 1 : integral 1 : answer (d) ()0Pt = when and () Entries are being processed most quickly at time 3 : () 1 : considers 0 1.}

9 Identifies candidates 1 : answer with justificationPt = t (hours) 0 2 5 7 8 ()Et (hundreds of entries) 0 4 13 21 23 AP Calculus AB 2010 Scoring Guidelines Question 3 2010 The College Board. Visit the College Board on the Web: There are 700 people in line for a popular amusement-park ride when the ride begins operation in the morning. Once it begins operation, the ride accepts passengers until the park closes 8 hours later.

10 While there is a line, people move onto the ride at a rate of 800 people per hour. The graph above shows the rate, (),rt at which people arrive at the ride throughout the day. Time t is measured in hours from the time the ride begins operation. (a) How many people arrive at the ride between 0t= and 3?t= Show the computations that lead to your answer. (b) Is the number of people waiting in line to get on the ride increasing or decreasing between 2t= and 3 ?t= Justify your answer.